3.17.54 \(\int \sqrt {1-2 x} (2+3 x)^3 (3+5 x) \, dx\)

Optimal. Leaf size=66 \[ -\frac {135}{176} (1-2 x)^{11/2}+\frac {69}{8} (1-2 x)^{9/2}-\frac {153}{4} (1-2 x)^{7/2}+\frac {3283}{40} (1-2 x)^{5/2}-\frac {3773}{48} (1-2 x)^{3/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} -\frac {135}{176} (1-2 x)^{11/2}+\frac {69}{8} (1-2 x)^{9/2}-\frac {153}{4} (1-2 x)^{7/2}+\frac {3283}{40} (1-2 x)^{5/2}-\frac {3773}{48} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x),x]

[Out]

(-3773*(1 - 2*x)^(3/2))/48 + (3283*(1 - 2*x)^(5/2))/40 - (153*(1 - 2*x)^(7/2))/4 + (69*(1 - 2*x)^(9/2))/8 - (1
35*(1 - 2*x)^(11/2))/176

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x) \, dx &=\int \left (\frac {3773}{16} \sqrt {1-2 x}-\frac {3283}{8} (1-2 x)^{3/2}+\frac {1071}{4} (1-2 x)^{5/2}-\frac {621}{8} (1-2 x)^{7/2}+\frac {135}{16} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {3773}{48} (1-2 x)^{3/2}+\frac {3283}{40} (1-2 x)^{5/2}-\frac {153}{4} (1-2 x)^{7/2}+\frac {69}{8} (1-2 x)^{9/2}-\frac {135}{176} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {1}{165} (1-2 x)^{3/2} \left (2025 x^4+7335 x^3+11205 x^2+9366 x+4442\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x),x]

[Out]

-1/165*((1 - 2*x)^(3/2)*(4442 + 9366*x + 11205*x^2 + 7335*x^3 + 2025*x^4))

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IntegrateAlgebraic [A]  time = 0.02, size = 60, normalized size = 0.91 \begin {gather*} \frac {-2025 (1-2 x)^{11/2}+22770 (1-2 x)^{9/2}-100980 (1-2 x)^{7/2}+216678 (1-2 x)^{5/2}-207515 (1-2 x)^{3/2}}{2640} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x),x]

[Out]

(-207515*(1 - 2*x)^(3/2) + 216678*(1 - 2*x)^(5/2) - 100980*(1 - 2*x)^(7/2) + 22770*(1 - 2*x)^(9/2) - 2025*(1 -
2*x)^(11/2))/2640

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fricas [A]  time = 1.17, size = 34, normalized size = 0.52 \begin {gather*} \frac {1}{165} \, {\left (4050 \, x^{5} + 12645 \, x^{4} + 15075 \, x^{3} + 7527 \, x^{2} - 482 \, x - 4442\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^3*(3+5*x)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/165*(4050*x^5 + 12645*x^4 + 15075*x^3 + 7527*x^2 - 482*x - 4442)*sqrt(-2*x + 1)

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giac [A]  time = 1.29, size = 74, normalized size = 1.12 \begin {gather*} \frac {135}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {69}{8} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {153}{4} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {3283}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {3773}{48} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^3*(3+5*x)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

135/176*(2*x - 1)^5*sqrt(-2*x + 1) + 69/8*(2*x - 1)^4*sqrt(-2*x + 1) + 153/4*(2*x - 1)^3*sqrt(-2*x + 1) + 3283
/40*(2*x - 1)^2*sqrt(-2*x + 1) - 3773/48*(-2*x + 1)^(3/2)

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maple [A]  time = 0.00, size = 30, normalized size = 0.45 \begin {gather*} -\frac {\left (2025 x^{4}+7335 x^{3}+11205 x^{2}+9366 x +4442\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{165} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^3*(5*x+3)*(-2*x+1)^(1/2),x)

[Out]

-1/165*(2025*x^4+7335*x^3+11205*x^2+9366*x+4442)*(-2*x+1)^(3/2)

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maxima [A]  time = 0.52, size = 46, normalized size = 0.70 \begin {gather*} -\frac {135}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {69}{8} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {153}{4} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {3283}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {3773}{48} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^3*(3+5*x)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-135/176*(-2*x + 1)^(11/2) + 69/8*(-2*x + 1)^(9/2) - 153/4*(-2*x + 1)^(7/2) + 3283/40*(-2*x + 1)^(5/2) - 3773/
48*(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 46, normalized size = 0.70 \begin {gather*} \frac {3283\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {3773\,{\left (1-2\,x\right )}^{3/2}}{48}-\frac {153\,{\left (1-2\,x\right )}^{7/2}}{4}+\frac {69\,{\left (1-2\,x\right )}^{9/2}}{8}-\frac {135\,{\left (1-2\,x\right )}^{11/2}}{176} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(3*x + 2)^3*(5*x + 3),x)

[Out]

(3283*(1 - 2*x)^(5/2))/40 - (3773*(1 - 2*x)^(3/2))/48 - (153*(1 - 2*x)^(7/2))/4 + (69*(1 - 2*x)^(9/2))/8 - (13
5*(1 - 2*x)^(11/2))/176

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sympy [A]  time = 2.73, size = 58, normalized size = 0.88 \begin {gather*} - \frac {135 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} + \frac {69 \left (1 - 2 x\right )^{\frac {9}{2}}}{8} - \frac {153 \left (1 - 2 x\right )^{\frac {7}{2}}}{4} + \frac {3283 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} - \frac {3773 \left (1 - 2 x\right )^{\frac {3}{2}}}{48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**3*(3+5*x)*(1-2*x)**(1/2),x)

[Out]

-135*(1 - 2*x)**(11/2)/176 + 69*(1 - 2*x)**(9/2)/8 - 153*(1 - 2*x)**(7/2)/4 + 3283*(1 - 2*x)**(5/2)/40 - 3773*
(1 - 2*x)**(3/2)/48

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